Heat content estimates over sets of finite perimeter

Valverde, Luis Acuña

doi:10.1016/j.jmaa.2016.03.087

Cited by 10 publications

(11 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First we consider the isotropic (rotationally invariant) α-stable process in R d . The following example shows that our theorems can be regarded as extensions of the results contained in papers [2] and [1].…”

Section: Theoremmentioning

confidence: 66%

“…where (A, γ, ν) is the triplet from (1). For Lévy processes with finite variation we have the following.…”

Section: Preliminariesmentioning

confidence: 99%

“…There are a lot of articles where bounds and asymptotic behaviour of the heat content related to Brownian motion, either on R d or on compact manifolds, were studied, see [19], [21], [22], [20], [18], [23]. Recently Acuña Valverde [2] investigated the heat content for isotropic stable processes in R d , see also [1] and [3]. In this paper we study the small time behaviour of the heat content associated with rather general Lévy processes in R d .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Heat content for convolution semigroups

Cygan

Grzywny

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Abstract. Let X = {X t } t≥0 be a Lévy process in R d and Ω be an open subset of R d with finite Lebesgue measure. In this article we consider the quantity H(t) = Ω P x (X t ∈ Ω c ) dx which is called the heat content. We study its asymptotic behaviour as t goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

show abstract

Section: Theoremmentioning

confidence: 66%

“…where (A, γ, ν) is the triplet from (1). For Lévy processes with finite variation we have the following.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Heat content for convolution semigroups

Cygan

Grzywny

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The heat content with respect to Lévy processes, especially Brownian motions, has been studied extensively, see, for instance, . The spectral heat content

Q_{D}^{(2)} false(t false)

with respect to Brownian motion has also been studied a lot (see ).…”

Section: Introductionmentioning

confidence: 99%

“…The heat content with respect to Lévy processes, especially Brownian motions, has been studied extensively, see, for instance, . The spectral heat content

Q_{D}^{(2)} false(t false)

with respect to Brownian motion has also been studied a lot (see ). In , a two‐term small time expansion for

Q_{D}^{(2)} false(t false)

was established for bounded

C^{1, 1}

domains and in a three‐term small time expansion for

Q_{D}^{(2)} false(t false)

was obtained for bounded domains with

C^{3}

boundary.…”

Section: Introductionmentioning

confidence: 99%

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α‐stable processes

Park

Song

2019

Bull. London Math. Soc.

View full text Add to dashboard Cite

In this paper, we study the small time asymptotic behavior of the spectral heat content Q (α) D (t) of an arbitrary bounded C 1,1 domain D with respect to the subordinate killed Brownian motion in D via an (α/2)-stable subordinator. For all α ∈ (0, 2), we establish a two-term small time expansion for Q(α) D (t) in all dimensions. When α ∈ (1, 2) and d 2, we establish a three-term small time expansion for Q(α) D (t).

show abstract

Spectral heat content for Lévy processes

Grzywny

Park

Song

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in Rd, d≥1. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in double-struckR with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

show abstract

Heat content estimates over sets of finite perimeter

Cited by 10 publications

References 21 publications

Heat content for convolution semigroups

Heat content for convolution semigroups

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α‐stable processes

Spectral heat content for Lévy processes

Contact Info

Product

Resources

About